In this post an overview of common problems and sollutions with e-mailboxes.
Transfer Email (Pesk, Cpanel, DirectAdmin)
imap server sync is a methode that can be used for all imap-enabled mailservers: https://imapsync.lamiral.info/
Webmail standard on CPanel
If your webhosting has a CPanel installed as part of their webhostingmanager (WHM) software, email configuration can be done locally (on your server) or virtually (domain bridge). For Cpanel local usage email can be used directly on the server with webmail: mail.domainname.ext or domainname.ext/webmail
For transfering between cpanel hosting see
Many users allready have an gmail account running on their devices, they use the gamail app or another applications to collect and send email.
It is possible to incorporate a non-gmail account in this application.
Email in gmail applications